The first SGM is noise conditional score network (NCSN) [1]}. Denoting the NCSN model as \(\mathbf {s}(\mathbf {x},\sigma )\) . It is trained to match the score function \(\bigtriangledown _{\mathbf {x}}\log p_{\sigma }(\mathbf {x})\) , where \(p_{\sigma }(\cdot ) = \mathcal {N}(\cdot ;\mathbf {x}^{\ast },\sigma ^2I_d)\)  is a Gaussian distribution with \(\sigma \)  the variance of the additive noises. As \(\sigma \)  increases from 0 to a large enough \(\sigma _{L}\) , \(p_{\sigma }(\cdot )\)  will transform gradually from the data distribution \(p_{\mathbf {x}^{\ast }}\)  to an approximately Gaussian distribution. Instead, the NCSN model \(\mathbf {s}(\mathbf {x},\sigma )\)  reverses the corruption process, starting from a Gaussian distribution and traveling to the target distribution \(p_{\mathbf {x}^{\ast }}\)  by an annealed Langevin dynamics.
[2]} interpreted SGMs as discrete versions of the various stochastic differential equations (SDEs).
In this perspective, diffusion sampling can be considered as a process in which we numerically solve a backward SDE. Based on this formula, [2]} proposed the NCSN++ model good at high-resolution image generation.
